Mister Exam
Factor -x^2-x+1 squared
The solution
Factorization
[src]
/ ___\ / ___\ | 1 \/ 5 | | 1 \/ 5 | |x + - - -----|*|x + - + -----| \ 2 2 / \ 2 2 /
$$\left(x + \left(\frac{1}{2} - \frac{\sqrt{5}}{2}\right)\right) \left(x + \left(\frac{1}{2} + \frac{\sqrt{5}}{2}\right)\right)$$
(x + 1/2 - sqrt(5)/2)*(x + 1/2 + sqrt(5)/2)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- x^{2} - x\right) + 1$$
To do this, let's use the formula
$$a x^{2} + b x + c = a \left(m + x\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = -1$$
$$b = -1$$
$$c = 1$$
Then
$$m = \frac{1}{2}$$
$$n = \frac{5}{4}$$
So,
$$\frac{5}{4} - \left(x + \frac{1}{2}\right)^{2}$$
$$\left(- x^{2} - x\right) + 1$$
To do this, let's use the formula
$$a x^{2} + b x + c = a \left(m + x\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = -1$$
$$b = -1$$
$$c = 1$$
Then
$$m = \frac{1}{2}$$
$$n = \frac{5}{4}$$
So,
$$\frac{5}{4} - \left(x + \frac{1}{2}\right)^{2}$$